Finite Group Databasebeta

← #20279699

Subgroups — #20279699

Order 6784.

540 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133216C212yesmaximal
2C2×#13133216C211yesmaximal, characteristic
3C2×#13133182C211yesmaximal, characteristic
4#16810555C211yesmaximal, characteristic
5#16810554C211yesmaximal, characteristic
6#16810553C211yesmaximal, characteristic
7#16810527C212yesmaximal
8#16810520C212yesmaximal
9#16810518C212yesmaximal
10#16810474C211yesmaximal, characteristic
11C53×#398C211yesmaximal, characteristic
12C22×Dic106C2212yes—
13C22×Dic106C2211yescharacteristic
14#13133216—24no—
15#13133216—28no—
16#13133182—24no—
17C2×C4⋊Dic53C2212yes—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2212yes—
25C2×C4×Dic53C2211yescharacteristic
26C2×C4×Dic53C2211yescharacteristic
27C2×C4×Dic53C2212yes—
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic
36#95×C53C2212yes—
37#95×C53C2211yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2312yes—
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C2×Dic106—22no—
50C2×Dic106—22no—
51C2×Dic106—24no—
52C2×Dic106—24no—
53C2×Dic106—24no—
54C2×Dic106D414yes—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—24no—
57C4⋊C4×C53—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53—24no—
60#11719919—22no—
61#11719919—24no—
62#11719919—22no—
63#11719919—24no—
64#11719919—24no—
65#11719919Q818yescomplemented
66C4×Dic53—22no—
67C4×Dic53—22no—
68C4×Dic53—24no—
69C4×Dic53D414yes—
70C23×C53C2411yescharacteristic
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53C2×D412yes—
82C2×C4×C53C2×D412yes—
83C2×C4×C53C2×Q814yes—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—21no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53C2×D412yes—
94C2×Dic53C2×D412yes—
95C2×Dic53C2×D412yes—
96C2×Dic53C2×D412yes—
97C2×Dic53C2×Q814yes—
98C2×Dic53C2×Q814yes—
99C2×Dic53#4112yes—
100Dic106—28no—
101Dic106—48no—
102C22×C53C22×D411yescharacteristic
103C22×C53C22×D411yescharacteristic
104C22×C53C22×Q812yes—
105C22×C53C2×#4111yescharacteristic
106C22×C532−1+412yes—
107C4×C53—22no—
108C4×C53—22no—
109C4×C53—24no—
110C4×C53—42no—
111C4×C53—41no—
112Dic53—22no—
113Dic53—24no—
114Dic53—42no—
115Dic53—41no—
116Dic53#12114yes—
117Dic53C4⋊Q814yescomplemented
118#2546—531nomaximal, Sylow
119C2×C53C2×#12111yescharacteristic
120C2×C53C2×C4⋊Q811yescharacteristic
121C2×C53#53611yescharacteristic
122C2×C53D4×Q812yes—
123C2×C53#55712yes—
124C2×C4⋊Q8—531no—
125C2×C4⋊Q8—532no—
126C2×C4×Q8—531no—
127#398—531no—
128#398—531no—
129#398—531no—
130#398D5311yescharacteristic
131#391—532no—
132#389—532no—
133#389—532no—
134C4×C4⋊C4—531no—
135C53#254611yescharacteristic, complemented, Sylow
136C22×Q8—531no—
137C22×Q8—532no—
138C4⋊Q8—1064no—
139C4⋊Q8—1068no—
140C4×Q8—1064no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—531no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4C2×D5311yescharacteristic
150C2×C4⋊C4C2×D5312yes—
151C2×C4⋊C4C2×D5311yescharacteristic
152C2×C42—531no—
153C2×C42—531no—
154C2×C42—531no—
155C2×C42—532no—
156#95—531no—
157#95—532no—
158#95—532no—
159#95—532no—
160#95C2×D5311yescharacteristic
161#95C2×D5312yes—
162C2×Q8—534no—
163C2×Q8—1062no—
164C2×Q8—1062no—
165C2×Q8—1064no—
166C2×Q8—1064no—
167C2×Q8—1064no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—532no—
171C22×C4—531no—
172C22×C4—531no—
173C22×C4—532no—
174C22×C4C22×D5311yescharacteristic
175C22×C4C22×D5312yes—
176C22×C4C22×D5311yescharacteristic
177C22×C4C22×D5311yescharacteristic
178C22×C4C22×D5312yes—
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—538no—
183C4⋊C4—1062no—
184C4⋊C4—1064no—
185C4⋊C4—1064no—
186C4⋊C4—1064no—
187C4⋊C4—1062no—
188C4⋊C4—1064no—
189C4⋊C4—1062no—
190C42—534no—
191C42—1062no—
192C42—1062no—
193C42—1064no—
194C23C23×D5311yescharacteristic
195Q8—1068no—
196Q8—2128no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—532no—
208C2×C4—532no—
209C2×C4—532no—
210C2×C4—534no—
211C2×C4—532no—
212C2×C4—532no—
213C2×C4—534no—
214C2×C4—1062no—
215C2×C4—1061no—
216C2×C4—1061no—
217C2×C4—1062no—
218C2×C4—1061no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4D4×D5312yes—
224C2×C4D4×D5312yes—
225C2×C4Q8×D5314yes—
226C22C2×D4×D5311yescharacteristic
227C22C2×D4×D5311yescharacteristic
228C22C2×Q8×D5312yes—
229C22#41×D5311yescharacteristic
230C22#1313326412yes—
231C4—24no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—534no—
237C4—534no—
238C4—1064no—
239C4—1062no—
240C4—2122no—
241C4—2121no—
242C2#121×D5311yescharacteristic
243C2C4⋊Q8×D5311yescharacteristic
244C2#1681119511yescharacteristic
245C2#1681128612yes—
246C2#1681135012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2546
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
65. #11719919Q8semidirect
117. Dic53C4⋊Q8semidirect
135. C53#2546semidirect